- #1
CAF123
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Homework Statement
Evaluate the surface integral [tex] \int_{S} \int \vec{F} \cdot \vec{n}\,dS[/tex]with the vector field [itex] \vec{F⃗}=zx\vec{i}+xy\vec{j}+yz\vec{k} [/itex]. S is the closed surface composed of a portion of the cylinder[itex] x^2 + y^2 = R^2 [/itex]that lies in the first octant, and portions of the planes x=0, y=0, z=0 and z=H. [itex] \vec{n} [/itex] is the outward unit normal vector.
2.Attempt at a solution
Attempt: I said S consisted of the five surfaces S1,S2,S3,S4 and S5. S1 being the portion of the cylinder, S2 being where the plane z=0 cuts the cylinder and similarly, S3,S4,S5 being where the planes x=0,y=0 and z=H cut the cylinder.
For S2, the normal vector points in the -k direction. so the required integral over S2 is: [tex]
\int_{0}^{R} \int_{0}^{\sqrt{R^2- x^2}} -yz\, dy\,dx [/tex]
Am I correct? I think for the surface S5 the only thing that would change in the above would be that the unit normal vector points in the positive k direction?
I need some guidance on how to set up the integrals for the rest of the surfaces.(excluding the cylinder part - I used cylindrical coords here and have an answer) I tried [itex] \int_{0}^{R} \int_{0}^{H} -xy\,dz\,dx [/itex] for the y = 0 plane intersection with the cylinder, but I am not sure if this is correct.
Any advice on how to tackle the remaining surfaces would be very helpful.